Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

Krysko, Anton V.; Awrejcewicz, Jan; Bodyagina, K. S.; Zhigalov, Maxim V.; Krysko, Vadim A.

doi:10.1007/s00707-021-03010-8

Cited by 5 publications

(5 citation statements)

References 50 publications

(54 reference statements)

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“…At each time step, an iterative procedure of the method of Birger’s variable elasticity parameters is applied [ 40 ]. This procedure was introduced in [ 57 , 59 ] for the calculation of FN metal plates in a three-dimensional formulation by the finite element method.…”

Section: Numeric Experiments and The Results Discussionmentioning

confidence: 99%

A New Mathematical Model of Functionally Graded Porous Euler–Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities

et al. 2022

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A new mathematical model of flexible physically (FN), geometrically (GN), and simultaneously physically and geometrically (PGN) nonlinear porous functionally graded (PFG) Euler–Bernoulli beams was developed using a modified couple stress theory. The ceramic phase of the functionally material was considered as an elastic material. The metal phase was considered as a physically non-linear material dependent on coordinates, time, and stress–strain state, which gave the opportunity to apply the deformation theory of plasticity. The governing equations of the beam as well as boundary and initial conditions were derived using Hamilton’s principle and the finite difference method (FDM) with a second-order approximation. The Cauchy problem was solved by several methods such as Runge–Kutta from 4-th to 8-th order accuracy and the Newmark method. Static problems, with the help of the establishment method, were solved. At each time step, nested iterative procedures of Birger method of variable elasticity parameters and Newton’s method were built. The Mises criterion was adopted as a criterion for plasticity. Three types of porosity-dependent material properties are incorporated into the mathematical modeling. For metal beams, taking into account the geometric and physical nonlinearity, the phenomenon of changing the boundary conditions, i.e., constructive nonlinearity (CN) was found.

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Section: Numeric Experiments and The Results Discussionmentioning

confidence: 99%

A New Mathematical Model of Functionally Graded Porous Euler–Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities

et al. 2022

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“…Although, in general, the dependence of the stress intensity on the strain intensity is usually determined experimentally, there are also available analytical expressions [31].…”

Section: Methods Of Variable Parameters Of Elasticity (3d Problems)mentioning

confidence: 99%

“…Krysko et al [30] presented mathematical models of physically nonlinear beams and plates in 3D and 1D formulation on the basis of the kinematic models of Euler-Bernoulli and Timoshenko. In addition, mathematical models for 3D physically nonlinear beams and plates fabricated from different modular materials were constructed [31]. It is well known and documented that various cutouts in plates reduce their mechanical strength and carrying load ability.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation

et al. 2021

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Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.

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“…Due to the nonlinearity of the problem, equation (9) is not the exact one. Substituting equation (9) into equation (7), the solution error for the first-order approximation is obtained as follows:…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Rods, beams and plates are important structural components that are utilized widely in most of the engineering structures such as buildings, industrial machinery, automobiles and aircrafts. These structures are modeled mathematically in the form of linear or nonlinear differential equations to investigate their mechanical behavior under various external loadings and boundary conditions [1][2][3][4][5][6][7][8][9]. Vibration analysis of some mechanical systems often leads to a nonlinear differential equation, similar to that of a nonlinear oscillator [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

Mohammadian,

Ismail

2024

Phys. Scr.

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The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

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Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials

Cited by 5 publications

References 50 publications

A New Mathematical Model of Functionally Graded Porous Euler–Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities

A New Mathematical Model of Functionally Graded Porous Euler–Bernoulli Nanoscaled Beams Taking into Account Some Types of Nonlinearities

Mathematical modeling of planar physically nonlinear inhomogeneous plates with rectangular cuts in the three-dimensional formulation

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

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